Update

6 files changed, 17 insertions, 22 deletions

diff --git a/pages/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
index 91906cd..0913776 100644
--- a/pages/functional-analysis-basics/banach-alaoglu-theorem.md
+++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
@@ -19,4 +19,6 @@ The {{ page.title }} is a special case of the following result:
 The polar of a neighborhood of zero in a locally convex space is weak\* compact.
 {% endtheorem %}
 
-See [Alaoglu–Bourbaki Theorem]({% link pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md %}) for more information.
+See
+[Alaoglu–Bourbaki Theorem](/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.html)
+for more information.
diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md
index 781fb1f..58ca1d3 100644
--- a/pages/functional-analysis-basics/reflexive-spaces.md
+++ b/pages/functional-analysis-basics/reflexive-spaces.md
@@ -17,7 +17,7 @@ $$
 where the functional $g_x$ on $X'$ is defined by
 
 $$
-g_x(f) = f(x) \quad \text{for $f \in X'$,} 
+g_x(f) = f(x) \quad \text{for $f \in X'$,}
 $$
 
 is called the *canonical embedding* of $X$ into its bidual $X''$.
@@ -79,7 +79,7 @@ C : X \longrightarrow X'', \quad C(x)(f) = f(x), \quad x \in X, f \in X',
 $$
 
 is an isomorphism.
-Therefore, the the dual map
+Therefore, the dual map
 
 $$
 C' : X''' \longrightarrow X', \quad C'(h)(x) = h(C(x)), \quad x \in X, h \in X''',
diff --git a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
index f6a9783..e0ec62b 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
@@ -33,10 +33,11 @@ This shows that $\graph{T}$ is closed.
 
 Conversely, suppose that $\graph{T}$ is a closed subspace of $X \times Y$.
 Note that $X \times Y$ is a Banach space with norm $\norm{(x,y)} = \norm{x} + \norm{y}$.
-Therefore $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$.
+Therefore, $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$.
 The canonical projections $\pi_X : \graph{T} \to X$ and $\pi_Y : \graph{T} \to Y$ are bounded.
-Clearly, $\pi_X$ is bijective, so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the
+Clearly, $\pi_X$ is bijective,
+so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the
 [Bounded Inverse Theorem](/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.html#bounded-inverse-theorem).
-Consequently the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded.
+Consequently, the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded.
 To complete the proof, observe that $\pi_Y \circ \pi_X^{-1} = T$.
 {% endproof %}
diff --git a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md
index 18cf64a..a2602ac 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md
@@ -6,7 +6,6 @@ nav_order: 1
 ---
 
 # {{ page.title }}
-{: .no_toc }
 
 In fact, there are multiple theorems and corollaries
 which bear the name Hahn–Banach.
@@ -14,15 +13,6 @@ All have in common that
 they guarantee the existence of linear functionals
 with various additional properties.
 
-<details open markdown="block">
-  <summary>
-    Table of contents
-  </summary>
-  {: .text-delta }
-- TOC
-{:toc}
-</details>
-
 {% definition Sublinear Functional %}
 A functional $p$ on a real vector space $X$
 is called *sublinear* if it is
@@ -146,7 +136,7 @@ we define a functional $f_0$ by $f_0(\alpha x) = \alpha \norm{x}$ for $\alpha \i
 It is easy to check that $f_0$ is linear and bounded with norm $\norm{f_0} = 1$.
 By the Hahn–Banach Extension Theorem for Normed Spaces,
 there exists a bounded linear functional $f$ on $X$ extending $f_0$ with identical norm.
-Hence we have $f(x) = f_0(x) = \norm{x}$ and $\norm{f} = \norm{f_0} = 1$.
+Hence, we have $f(x) = f_0(x) = \norm{x}$ and $\norm{f} = \norm{f_0} = 1$.
 {% endproof %}
 
 Recall that for a normed space $X$ we denote its (topological) dual space by $X'$.
@@ -155,7 +145,7 @@ Recall that for a normed space $X$ we denote its (topological) dual space by $X'
 For every element $x$ of a real or complex normed space $X$ one has
 
 $$
-\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} 
+\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}}
 $$
 
 and the supremum is attained.
diff --git a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md
index b191bb2..e7f2b70 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md
@@ -30,10 +30,10 @@ This remains true, if we take closures:
 $\bigcup \overline{mTB_X} = Y$.
 Hence, we have written the space $Y$,
 which is assumed to have a complete norm,
-as the union of countably many closed sets. It follows form the
+as the union of countably many closed sets. It follows from the
 [Baire Category Theorem]({% link pages/general-topology/baire-spaces.md %})
 that $\overline{mTB_X}$ has nonempty interior for some $m$.
-Thus there are $q \in Y$ and $\alpha > 0$
+Thus, there are $q \in Y$ and $\alpha > 0$
 such that $q + \alpha B_Y \subset \overline{mTB_X}$.
 Choose a $p \in X$ with $Tp=q$.
 It is a well known fact, that in a normed space
diff --git a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
index 47ddd3f..1140e45 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
@@ -14,11 +14,13 @@ Let $X$, $Y$ be normed spaces.
 We say that a collection $\mathcal{T}$ of bounded linear operators
 from $X$ to $Y$ is
 {: .mb-0 }
-- *pointwise bounded* if the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded for every $x \in X$,
+- *pointwise bounded* if the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded
+  for every $x \in X$,
 - *uniformly bounded* if the set $\braces{\norm{T} : T \in \mathcal{T}}$ is bounded.
 {% enddefinition %}
 
-Clearly, every uniformly bounded collection of operators is pointwise bounded since $\norm{Tx} \le \norm{T} \norm{x}$.
+Clearly, every uniformly bounded collection of operators is pointwise bounded
+since $\norm{Tx} \le \norm{T} \norm{x}$.
 The converse is true, if $X$ is complete:
 
 {% theorem * Uniform Boundedness Theorem %}